Download The Unit Circle

Introduction
A unit circle is a circle that has a radius of 1 unit, with
the center of the circle located at the origin of the
coordinate plane. Because r = 1 in the unit circle, it can
be a useful tool for discussing arc lengths and angles in
circles. An angle in a unit circle can be studied in radians
or degrees; however, since radians directly relate an
angle measure to an arc length, radian measures are
more useful in calculations.
1
5.1.2: The Unit Circle
Key Concepts
• Angles are typically in standard position on a unit
circle. This means that the center of the circle is
placed at the origin of the coordinate plane, and the
vertex of the angle is on the origin at the center of the
circle. The initial side of the angle (the stationary ray
from which the measurement of the angle starts) is
located along the positive x-axis. The terminal side
(the movable ray that determines the measure of the
angle) may be in any location.
2
5.1.2: The Unit Circle
Key Concepts, continued
3
5.1.2: The Unit Circle
Key Concepts, continued
• The terminal side of the angle may be rotated
counterclockwise to create a positive angle or
clockwise to create a negative angle.
• To sketch an angle in radians on the unit circle,
remember that halfway around the circle (180°) is
equal to  radians and that a full rotation (360°) is
equal to 2 radians. Then use the fraction of  to
estimate the angle’s location, if it falls somewhere
between these measures.
4
5.1.2: The Unit Circle
Key Concepts, continued
•
Within the unit circle, each angle has a reference
angle. The reference angle is always the angle that
the terminal side makes with the x-axis. The
reference angle’s sine, cosine, and tangent are the
same as that of the original angle except for the sign,
which is based on the quadrant in which the terminal
side is located.
5
5.1.2: The Unit Circle
Key Concepts, continued
•
Recall that a right triangle has one right angle and
two acute angles (less than 90°). Sine, cosine, and
tangent are trigonometric functions of an acute
angle  in a right triangle and are determined by the
ratios of the lengths of the opposite side, adjacent
side, and the hypotenuse of that triangle,
summarized as follows.
6
5.1.2: The Unit Circle
Key Concepts, continued
• The sine of  = sin  =
length of opposite side
length of hypotenuse
• The cosine of  = cos  =
.
length of adjacent side
• The tangent of  = tan  =
length of hypotenuse
.
length of opposite side
length of adjacent side
.
7
5.1.2: The Unit Circle
Key Concepts, continued
• To find a reference angle, first sketch the original
angle to determine which quadrant it lies in. Then,
determine the measure of the angle between the
terminal side and the x-axis. The following table
shows the relationships between the reference angle
and the original angle ( ) for each quadrant.
Quadrant
Reference angle
(degrees)
Reference angle
(radians)
I
same as 
same as 
II
180° – 
III
 – 180°
360° – 
 radians – 
 –  radians
2 radians – 
IV
5.1.2: The Unit Circle
8
Key Concepts, continued
• If an angle is larger than 2 radians (360°), subtract a
full rotation (2 radians or 360°) until the angle is less
than 2 radians (360°). Then, find the reference angle
of the resulting angle.
• The coordinates of the point at which the terminal side
intersects the unit circle are always given by (cos  ,
sin  ), where  is the measure of the angle.
9
5.1.2: The Unit Circle
Common Errors/Misconceptions
• mistakenly thinking that the reference angle in
Quadrants II and IV is the angle that the terminal side
makes with the y-axis
• reversing sin  and cos  when attempting to find the
coordinates of the point where the terminal side
intersects the unit circle
10
5.1.2: The Unit Circle
Guided Practice
Example 1
2p
On a unit circle, sketch angles that measure
radians,
3
p
9p
radian, and
radians.
4
7
11
5.1.2: The Unit Circle
Guided Practice: Example 1, continued
1. Sketch a unit circle, then label  radians
and 2 radians.
A half rotation (180°) is  radians and a full rotation
(360°) is 2 radians. Notice that 0 radians and
2 radians are in the same location on the unit circle,
but represent different angle measures.
12
5.1.2: The Unit Circle
Guided Practice: Example 1, continued
13
5.1.2: The Unit Circle
Guided Practice: Example 1, continued
2p
2. Sketch
radians.
3
2p
is the same as
2
p. In other words, the terminal
3
3
2
side is of the way between 0 and . Thus, imagine
3
the semicircle between 0 radians and  radians split
into thirds, and then sketch the angle
2
3
of the way
around the semicircle.
14
5.1.2: The Unit Circle
Guided Practice: Example 1, continued
15
5.1.2: The Unit Circle
Guided Practice: Example 1, continued
3. Sketch
p
p
4
radian.
is the same as
1
p. In other words, it is
1
of the
4
4
4
way to . Thus, imagine the semicircle between
0 radians and  radians split into fourths, and then
sketch the angle
1
4
of the way around the
semicircle.
16
5.1.2: The Unit Circle
Guided Practice: Example 1, continued
17
5.1.2: The Unit Circle
Guided Practice: Example 1, continued
9p
4. Sketch
radians.
7
9p
2
which
is
equal
to
p,
1 p.
7
7
7
Because this value is greater than , it goes beyond
2
 radians. It is of the way past .
7
Imagine the semicircle between  radians and 2
is the same as
9
radians split into sevenths, and then sketch the angle
2
of the way around the semicircle.
7
5.1.2: The Unit Circle
18
Guided Practice: Example 1, continued
19
5.1.2: The Unit Circle
Guided Practice: Example 1, continued
5. Summarize your findings.
The diagram on the next slide shows the final unit
circle with angles that measure
and
9p
2p
3
radians,
p
4
radian,
radians.
7
20
5.1.2: The Unit Circle
Guided Practice: Example 1, continued
✔
21
5.1.2: The Unit Circle
Guided Practice: Example 1, continued
22
5.1.2: The Unit Circle
Guided Practice
Example 2
Find the reference angles for angles that measure
11p
3p
radians,
radians, and 5.895 radians.
9
5
23
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
11p
1. Sketch an angle with a measure of
9
radians on the unit circle.
11p
2
radians is the same as 1 p radians; therefore,
9
9
2
this angle will be of the way between  radians and
9
2 radians.
24
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
25
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
2. Determine the measure of the angle
between the terminal side and the x-axis.
Since the terminal side falls in Quadrant III, subtract
11p
 radians from the original angle measure,
radians,
9
to find the measure of the reference angle.
26
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
11p
9
=
=
Subtract  from the original
angle measure.
-p
11p
9
2p
-
9p
9
9
Rewrite  as a fraction with a
common denominator.
Subtract.
The reference angle for
11p
9
radians is
2p
9
radians.
27
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
3p
3. Sketch
radians.
5
Sketch
3p
5
 radians.
radians
3
5
of the way between 0 and
28
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
29
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
4. Determine the measure of the angle
between the terminal side and the x-axis.
Since the terminal side falls in Quadrant II, subtract
3p
radians from  radians to find the measure of the
5
reference angle.
30
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
p=
=
3p
5
5p
5
2p
Subtract the original angle
measure from .
-
3p
5
5
Rewrite  as a fraction with a
common denominator.
Subtract.
The reference angle for
3p
5
radians is
2p
5
radians.
31
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
5. Sketch 5.895 radians.
Since  is not included in this measurement, we
must use decimal approximations.  is
approximately 3.14 and 2 is approximately 6.28.
5.895 is fairly close to 6.28 and thus will fall in
Quadrant IV.
32
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
33
5.1.2: The Unit Circle
Guided Practice: Example 2, continued
6. Determine the measure of the angle
between the terminal side and the x-axis.
Since the terminal side falls in Quadrant IV, subtract
5.895 radians from 2 radians to find a more precise
measure of the reference angle.
2 – 5.895 ≈ 0.388
The measure of the reference angle for 5.895
radians is approximately 0.388 radian.
✔
5.1.2: The Unit Circle
34
Guided Practice: Example 2, continued
Applet A
Applet B
35
5.1.2: The Unit Circle